Question Number 180398 by a.lgnaoui last updated on 12/Nov/22
$$\int\frac{\mathrm{1}}{\mathrm{sin}\:\left(\mathrm{x}−\mathrm{a}\right)\mathrm{sin}\:\left(\mathrm{x}−\mathrm{b}\right)}\mathrm{dx}=? \\ $$$$ \\ $$
Commented by Spillover last updated on 12/Nov/22
https://youtu.be/J0cicATsvxM
Answered by MJS_new last updated on 12/Nov/22
$$\int\frac{{dx}}{\mathrm{sin}\:\left({x}−{a}\right)\:\mathrm{sin}\:\left({x}−{b}\right)}= \\ $$$$=\frac{\mathrm{1}}{\mathrm{sin}\:\left({a}−{b}\right)}\int\left(\mathrm{cot}\:\left({x}−{a}\right)\:−\mathrm{cot}\:\left({x}−{b}\right)\right){dx}= \\ $$$$=\frac{\mathrm{1}}{\mathrm{sin}\:\left({a}−{b}\right)}\left(\mathrm{ln}\:\mid\mathrm{sin}\:\left({x}−{a}\right)\mid\:−\mathrm{ln}\:\mid\mathrm{sin}\:\left({x}−{b}\right)\mid\right)= \\ $$$$=\frac{\mathrm{1}}{\mathrm{sin}\:\left({a}−{b}\right)}\mathrm{ln}\:\mid\frac{\mathrm{sin}\:\left({x}−{a}\right)}{\mathrm{sin}\:\left({x}−{b}\right)}\mid\:+{C} \\ $$